One way to extend Eq. (REF ) to the laboratory frame is to transform the magnetic field and the time to the laboratory frame, and include the Thomas correction [1]}, [2]}.
Another commonly used way is based on the explicitly covariant approach [3]} which is analyzed in detail in Refs. [4]}, [5]}. The corresponding equations can be written as
\(&& \frac{d \vec{\xi }}{d t} = \vec{\omega } \times \vec{\xi }, \\&& \vec{\omega } = \vec{\omega }_{\vec{H}} + \vec{\omega }_{\vec{E}}, \nonumber \\&& \vec{\omega }_{\vec{H}} = - \frac{e }{m } \left[ \left( \frac{g}{2} -1 + \frac{1}{\gamma } \right) \, \vec{H} - \left( \frac{g}{2}-1 \right) \, \frac{\gamma }{1+\gamma } \, \vec{v} \, (\vec{H} \, \vec{v} ) \right], \nonumber \\&& \vec{\omega }_{\vec{E}} = - \frac{e }{m } \left( \frac{g}{2}- \frac{\gamma }{1+\gamma } \right) \, \vec{E} \times \vec{v}, \nonumber \) 
